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 Chapter 11.1 Puzzle
This assignment is due 48 hours after the Chapter Test is
given.
- Puzzle solutions are due by 5:00 PM two days after the Chapter
Test.
- Puzzles are worth 10 points of extra credit each.
- Answers must be accompanied by valid reasoning. Just like
the tests, the answer alone isn't enough!
- Please enter your solution in the text area at the bottom
of this page. DON'T FORGET TO GIVE YOUR NAME!
"Families of
Hypocycloids"
(You won't find these
at the zoo.)
There is an interesting history behind epicycloids. After
Copernicus showed that the sun didn't move around the Earth,
astronomers believed that the planets moved in circular paths
around the Sun. Gradually, mathematical analysis showed that
this wasn't quite right. So, they posited that the "circular
paths" were actually epicycles: small circles rolling around
larger ones (See Stewart, p. 682). More accurate numerical data
showed that this theory was also wrong. It was then believed
that the paths were double-epicycles: circles rolling around
circles rolling around circles. Finally, Kepler (using Brahe's
data) showed that the paths were elliptical, and then Newton,
using his newly developed calculus, derived laws to show why
Kepler's discovery was true.
If a circle C rolls on the outside of the fixed
circle, the curve traced out by point P (a point on the circle
C) is called an epicycloid. Find the parametric equation for
the epicycloid. (See Stewart, p. 682, for additional information
about hypocycloids)
Don't forget to explain your answer
as clearly as possible.
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